Matrix Computations by Monte Carlo Optimization.
Abstract
This research addresses the general problem of performing matrix computations by Monte Carlo optimization. Four problems are examined: (1) estimating the value of a matrix function having a matrix power series representation, (2) estimating the solution of a system of linear equations, (3) estimating eigenvalues and eigenvectors of a matrix, and (4) estimating the condition number, determinant, and 2-norms of a matrix. To estimate the value of F(A) of a matrix function F having a convergent matrix power series, we derive and analyze unbiased consistent estimators of a matrix polynomial approximating the power series. A notion of an absolutely-convergent matrix power series is proposed and it is shown that if F has an absolutely-convergent matrix power series at A, then there exists an unbiased consistent estimator of F, thus eliminating the need for polynomial approximation. The methods may be applied to solve certain systems of differential equations. The Laplace method of approximating definite integrals is used to derive random variables for estimating the solution of a system of linear equations. They depend on a parameter which, if sufficiently large, yields an unbiased consistent estimator. Two methods are studied for sampling from the underlying population. The first uses Monte Carlo simulation of Markov processes. The second uses quasi-Monte Carlo methods adaptable to parallel computation.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1986
- Accession Number
- ADA176555
Entities
People
- Jack J. Murphy
Organizations
- Air Force Institute of Technology